The teacher’s death, which authorities say was violent, still is surrounded by questions about where and how she was killed, and why.

Still, investigators “have no reason to think this was anything other than an isolated attack by a stranger,” Mulroy told reporters Tuesday.

As the investigation unfolds, Fletcher’s community is mourning the junior kindergarten teacher and mother of two.

“We are heartbroken and devastated by this senseless loss. Liza was a such a joy to so many — her family, friends, colleagues, students, parents, members of her Second Presbyterian Church congregation, and everyone who knew her,” the statement reads.

“Now it’s time to remember and celebrate how special she was and to support those who cared so much for her. We appreciate all the expressions of love and concern we have received. We are grateful beyond measure to local, state and federal law enforcement for their tireless efforts to find Liza and to bring justice to the person responsible for this horrible crime,” Fletcher’s family said.

“We are heartbroken at the loss of our beloved teacher, colleague, and friend Liza Fletcher,” the St. Mary’s Episcopal School said.

Abston, 38, was arraigned Tuesday on his initial charges: Especially aggravated kidnapping and tampering with evidence. He was being held in a county jail on those charges with bond set at $510,000.

A judge Tuesday appointed a public defender to represent him after he told the court he didn’t have an attorney or the money to post bond.

Abston is set to be arraigned Wednesday on charges of first-degree murder, premeditated murder and murder in the course of the perpetration of a kidnapping, Mulroy said.

How the four-day search for the victim unfolded

Fletcher had been jogging in a neighborhood near the University of Memphis around 4 a.m. Friday. Her husband told police Friday morning that she’d not returned, authorities said in an affidavit, which initially was filed Sunday and amended Tuesday.

Someone found her phone in a street that morning, and it was given to one of Fletcher’s relatives, who gave it to investigators, the affidavit reads.

Police then found surveillance video of that area, which shows a black GMC Terrain pass by her, according to the affidavit. A man is seen in the footage getting out of the SUV and “aggressively” running toward her before forcing her into the vehicle’s passenger seat, according to the affidavit.

The SUV remained in a parking lot for about four minutes after both people were inside and then drove away, the affidavit states.

Police also analyzed a pair of sandals that were found at the abduction site, near the victim’s phone. DNA found on the shoes matched Abston’s DNA, the affidavit reads.

Investigators interviewed Abston’s employer, who said he drove a GMC Terrain and verified his phone number. Investigators checked Abston’s cell phone records, which showed he was near the abduction scene during the time of Fletcher’s kidnapping, according to the affidavit.

Members of a US Marshals task force found a GMC Terrain near Abston’s home on Saturday morning — and it had the same distinguishable damage seen in the surveillance footage, and the license plate matched the partial plate information gleaned from the video, the affidavit reads.

The task force detained Abston near his home Saturday, the court document said.

Police gathered details from two witnesses — including Abston’s brother — who say they saw him acting strangely at the brother’s Memphis house after the abduction, according to the affidavit.

Both said Abston cleaned the interior of the GMC Terrain with floor cleaner, and that he washed his clothes in the sink of the home, according to the affidavit.

Body found near vacant home; clothing found discarded nearby

On Monday, searchers, acting on information from an FBI team that analyzes cell phone data, found Fletcher’s body just after 5 p.m., according to the affidavit.

The searchers, using the cell phone data, had focused on an area near an intersection less than a mile from the brother’s home. The team smelled an odor of decay coming from an area near a vacant home, saw vehicle tracks in the grass near the driveway and eventually found a body on the property, according to the affidavit.

“The scene investigation revealed that the female fit the description of missing person, Eliza Fletcher,” the affidavit states.

That property was in the 1600 block of Victor Street, authorities said. That’s about a half-mile drive from the address that authorities gave for his brother’s home, a 7.5-mile drive from the alleged abduction site and about a 15.5-mile drive from Abston’s home.

A little more than an hour after the body was found and about three tenths of a mile away, a detective found a trash bag with purple running shorts — consistent with the ones Fletcher was wearing when she was abducted, the affidavit reads.

Suspect also faces charges not related to Fletcher

Abston is also facing charges unrelated to Fletcher’s case, including identity theft, theft of property $1,000 or less and fraudulent use/illegal possession of a credit or debit card $1,000 or less, Shelby County jail records show.

Those charges are connected to a theft report filed last week by a woman who reported someone was using her Cash App card and Wisely Card at gas stations without her knowledge.

CNN has reached out to the Shelby County district attorney and Memphis police regarding the theft charges.